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Four Ships

Puzzle

Four ships are sailing on a 2D planet. Each ships traverses a straight line at constant speed. No two ships are traveling parallel to each other. Their journeys started at some time in the distant past. Sometimes, a pair of ships collides. A ship continues its journey even after a collision. However, it is strong enough only to survive two collisions; it dies when it collides a third time. The situation is grim. Five of six possible collisions have already taken place (no collision involved more than 2 ships) and two ships are out of commission. What fate awaits the remaining two?

Source

I first heard this puzzle from Sanjeeb Dash in 1994 when we were together at IIT Delhi. Sanjeeb is presently at IBM Research.

Solution

Let z-axis denote time. Let x- and y- axes denote the 2D planet. Then the four trajectories are straight lines. Since no collision involved more than two ships, these four lines must all lie in a plane. So the sixth collision is certain.

Previous Puzzle: Fifteen Sum

Alice and Bob take turns to pick numbers from 1 thru 9, without replacement. The first to possess three distinct numbers that sum to 15 wins. Does Alice have a winning strategy?

Next Puzzle: Cap Colors

An evil troll once captured a bunch of gnomes and told them, "Tomorrow, I will make you stand in a file, ordered by height such that a gnome can see exactly those gnomes that are shorter than him. I will place either a white cap or a black cap on each head. Then, starting from the tallest, each gnome has to declare aloud what he thinks the color of his own cap is. In the end, those who were correct will be spared; others will be eaten, silently." The gnomes set thinking and came up with a strategy. How many of them survived?

12 Sep 2008
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